Assignment 3

1 General information

This assignment is related to Lecture 3 and BDA3 Chapters 2 and 3. Use Frank Harrell’s recommendations on how to state results in Bayesian two group comparisons (and note that there is no point null hypothesis testing in this assignment).

The maximum amount of points from this assignment is 9.

We have prepared two quarto templates specific to this assignment to help you get started:

• A recommended template (html, qmd, pdf) which uses some additional packages, which however requires a bit more set-up work to run and
• a simple template (html, qmd, pdf) which doesn’t use those additional packages and is therefore easier to get to run.

Tip

Reading instructions:

• The reading instructions for BDA3 Chapter 1.
• The reading instructions for BDA3 Chapter 2.

Grading instructions:

The grading will be done in peergrade. All grading questions and evaluations for this assignment are contained within this document in the collapsible Rubric blocks.

Further information

• The recommended tool in this course is R (with the IDE RStudio).
• Instead of installing R and RStudio on you own computer, see how to use R and RStudio remotely.
• If you want to install R and RStudio locally, download R and RStudio.
• There are tons of tutorials, videos and introductions to R and RStudio online. You can find some initial hints from RStudio Education pages.
• When working with R, we recommend writing the report using quarto and the provided template. The template includes the formatting instructions and how to include code and figures.
• Instead of quarto, you can use other software to make the PDF report, but the the same instructions for formatting should be used.
• Report all results in a single, anonymous *.pdf -file and submit it in peergrade.io.
• The course has its own R package aaltobda with data and functionality to simplify coding. The package is pre-installed in JupyterHub. To install the package on your own system, run the following code (upgrade="never" skips question about updating other packages):

install.packages("remotes")
remotes::install_github("avehtari/BDA_course_Aalto", subdir = "rpackage", upgrade="never")

• Many of the exercises can be checked automatically using the R package markmyassignment (pre-installed in JupyterHub). Information on how to install and use the package can be found in the markmyassignment documentation. There is no need to include markmyassignment results in the report.
• Recommended additional self study exercises for each chapter in BDA3 are listed in the course web page. These will help to gain deeper understanding of the topic.
• Common questions and answers regarding installation and technical problems can be found in Frequently Asked Questions (FAQ).
• Deadlines for all assignments can be found on the course web page and in Peergrade. You can set email alerts for the deadlines in Peergrade settings.
• You are allowed to discuss assignments with your friends, but it is not allowed to copy solutions directly from other students or from internet.
• You can copy, e.g., plotting code from the course demos, but really try to solve the actual assignment problems with your own code and explanations.
• Do not share your answers publicly.
• Do not copy answers from the internet or from previous years. We compare the answers to the answers from previous years and to the answers from other students this year.
• Use of AI is allowed on the course, but the most of the work needs to by the student, and you need to report whether you used AI and in which way you used them (See points 5 and 6 in Aalto guidelines for use of AI in teaching).
• All suspected plagiarism will be reported and investigated. See more about the Aalto University Code of Academic Integrity and Handling Violations Thereof.
• Do not submit empty PDFs, almost empty PDFs, copy of the questions, nonsense generated by yourself or AI, as these are just harming the other students as they can’t do peergrading for the empty or nonsense submissions. Violations of this rule will be reported and investigated in the same way was plagiarism.
• If you have any suggestions or improvements to the course material, please post in the course chat feedback channel, create an issue, or submit a pull request to the public repository!

Rubric S1: General information - 7.5/100 points

• Q1: Can you open the PDF and it’s not blank nor nonsense?
  
• Q2: Is the report anonymous?

2 Inference for normal mean and deviation (3 points)

A factory has a production line for manufacturing car windshields. A sample of windshields has been taken for testing hardness. The observed hardness values \(\mathbf{y}_1\) can be found in the dataset windshieldy1 in the aaltobda package.

We may assume that the observations follow a normal distribution with an unknown standard deviation \(\sigma\). We wish to obtain information about the unknown average hardness \(\mu\). For simplicity we assume standard uninformative prior discussed in the book, that is, \(p(\mu, \sigma) \propto \sigma^{-1}\). It is not necessary to derive the posterior distribution in the report, as it has already been done in the book (see section 3.2).

Subtask 2.a)

Formulate

1. the likelihood,
2. the prior, and
3. the resulting posterior.

Subtask 2.b)

What can you say about the unknown \(\mu\)?

• Compute and report the point estimate \(E(\mu|y)\),
• compute and report a posterior 95%-interval,
• plot the density, and
• write interpretation of the result in text.

Tip

Hint:

Posterior intervals are also called credible intervals and are different from confidence intervals.

Subtask 2.c)

What can you say about the hardness of the next windshield coming from the production line before actually measuring the hardness?

• Compute and report the point estimate \(E(\tilde{y}|y)\),
• compute and report a posterior predictive 95%-interval,
• plot the density, and
• write interpretation of the result in text.

Tip

Hint:

Predictive intervals are different from posterior intervals.

With a conjugate prior a closed form posterior is Student’s \(t\) form (see equations in the book).

Rubric S2: Inference for normal mean and deviation (3 points)

• Q3: Is the source code included?
  • No
  • Yes
• Q4: Are the likelihood, prior and the posterior for computing the average hardness value reported? It is ok to refer to the book instead of deriving the distributions.
  • No
  • Yes, but some are missing
  • Yes
• Q5: In part b), were the point estimates and posterior interval provided? (The posterior mean should be close to … and 95% posterior interval should be around …)
  • No
  • Yes, but seem incorrect or only one estimate was reported
  • Yes, and the reported values seem plausible
• Q6: In part b), was the density plotted?
  • No
  • Yes, but seem incorrect
  • Yes, and the plot seems plausible
• Q7: For c)-part, was a formula or a simulation method presented for computing the posterior predictive distribution? It is ok to refer to the book.
  • No
  • Yes, but seems incorrect
  • Yes
• Q8: For c)-part, were the point estimate and predictive interval provided? (95% predictive interval should be around … and the mean the same as in a)-part).
  • No
  • Yes, but seems incorrect
  • Yes, and the reported values seem plausible
• Q9: For c)-part, was the density plotted?
  • No
  • Yes, but seems incorrect
  • Yes, and the plot seem plausible

3 Inference for the difference between proportions (3 points)

An experiment was performed to estimate the effect of beta-blockers on mortality of cardiac patients. A group of patients was randomly assigned to treatment and control groups: out of 674 patients receiving the control, 39 died, and out of 680 receiving the treatment, 22 died. Assume that the outcomes are independent and binomially distributed, with probabilities of death of \(p_0\) and \(p_1\) under the control and treatment, respectively. Set up a noninformative or weakly informative prior distribution on \((p_0,p_1)\).

Subtask 3.a)

Formulate

1. the likelihood,
2. the prior, and
3. the resulting posterior.

Subtask 3.b)

Summarize the posterior distribution for the odds ratio, \[ \mathrm{OR} = (p_1/(1-p_1))/(p_0/(1-p_0)). \]

• Compute and report the point estimate \(E(\mathrm{OR}|y_0,y_1)\),
• compute and report a posterior 95%-interval,
• plot the histogram, and
• write interpretation of the result in text.

Use Frank Harrell’s recommendations how to state results in Bayesian two group comparison.

Tip

Hint

With a conjugate prior, a closed-form posterior is the Beta form for each group separately (see equations in the book). You can use rbeta() to sample from the posterior distributions of \(p_0\) and \(p_1\), and use this sample and odds ratio equation to get a sample from the distribution of the odds ratio.

Subtask 3.c)

Use at least two different priors, and discuss the sensitivity of your inference to your choice of prior density with a couple of sentences.

Rubric S3: Inference for the difference between proportions (3 points)

• Q10: Is the source code included?
  • No
  • Yes
• Q11: Are the likelihood, prior and the posterior for the death probabilities reported? It is ok to refer to the book instead of deriving the distributions.
  • No
  • Yes, but some are missing
  • Yes
• Q12: In part a), was the simulation algorithm for computing the posterior of the odds ratio presented or implemented?
  • No
  • Yes, but seems incorrect
  • Yes
• Q13: In part a), was the odds ratio summarized with a point estimate and a posterior interval? (The mean should be close to … and 95% posterior interval approximately …)
  • No
  • Yes, but results seem incorrect
  • Yes, and the results seem plausible
• Q14: In part b), was some discussion about testing alternative priors provided? (For example, one could have repeated the computations in a)-part which a couple of alternative priors and reported these results or some related general conclusions briefly)
  • Not at all
  • Some analysis was provided but it was lacking or did not make sense
  • Some alternative priors were tested and some sensible discussion provided

4 Inference for the difference between normal means (3 points)

Consider a case where the same factory has two production lines for manufacturing car windshields. Independent samples from the two production lines were tested for hardness. The hardness measurements for the two samples \(\mathbf{y}_1\) and \(\mathbf{y}_2\) be found in the datasets windshieldy1 and windshieldy2 in the aaltobda package.

We assume that the samples have unknown standard deviations \(\sigma_1\) and \(\sigma_2\). Use uninformative or weakly informative priors and answer the following questions:

Subtask 4.a)

Formulate

1. the likelihood,
2. the prior, and
3. the resulting posterior.

Subtask 4.b)

What can you say about \(\mu_d = \mu_1 - \mu_2\)?

• Compute and report the point estimate \(E(\mu_d|y_1, y_2)\),
• compute and report a posterior 95%-interval,
• plot the histogram, and
• write interpretation of the result in text.

Use Frank Harrell’s recommendations how to state results in Bayesian two group comparison.

Tip

Hint

With a conjugate prior, a closed-form posterior is Student’s \(t\) form for each group separately (see equations in the book). You can use the rtnew() function to sample from the posterior distributions of \(\mu_1\) and \(\mu_2\), and use this sample to get a sample from the distribution of the difference \(\mu_d = \mu_1 - \mu_2\).

Subtask 4.c)

Given this specific model, what is the probability that the means are exactly the same (\(\mu_1 = \mu_2\))? Explain your reasoning.

Rubric S4: Inference for the difference between normal means (3 points)

• Q15: Is source code included?
  • No
  • Yes
• Q16: Are the likelihood, prior and the posterior for the windshield hardness values reported? (It is also ok to refer to the book or related formulas from exercise 1)
  • No
  • Yes, but some are missing
  • Yes
• Q17: In part a), was the simulation algorithm for computing the difference in the means presented or implemented?
  • No
  • Yes, but seems to be incorrect
  • Yes
• Q18: In part a), was the posterior for the difference between the means summarized with point and interval estimates? (The mean should be close to … and 95% posterior interval … or something close to it)
  • No answer
  • Yes, but results seem incorrect or only one estimate was given
  • Yes, and results seem reasonable
• Q19: Were some analysis or discussion provided for assessing whether the means could be the same? …
  • No analysis or explanation is given
  • Yes, but the analysis or explanation seems incorrect
  • Yes, and the analysis or explanation seems plausible

5 Overall quality of the report

Rubric S5: Overall quality of the report - 7.5/100 points

• Q20: Does the report include whether AI was used and if was used explanation on how it was used?
  • No
  • Yes
• Q21: Does the report follow the formatting instructions?
  • Not at all
  • Little
  • Mostly
  • Yes
• Q22: In case the report doesn’t fully follow the formatting instructions, specify the formatting instruction that is not followed. If applicable, specify the page of the report, where this difference in formatting is visible.
• Q23: Please provide also feedback on the presentation (e.g. text, layout, flow of the responses, figures, figure captions). Part of the course is practicing making data analysis reports. By providing feedback on the report presentation and other students can learn what they can improve or what they already did well. You should be able to provide constructive or positive feedback for all non-empty reports. If you think the report is perfect, and you can’t come up with any suggestions how to improve, you can provide feedback on what you liked and why you think the report is better than yours.